Frouded Number and Waterwave Speed

[dbharrison1]

I have a query about the relationship between Froude Number and water wave propogation velocity in the field of hull design.

Froude Number is Flow Velocity/Water Wave propogation velocity.
Where Flow velocity is speed of vessel V.

Water Wave Propogation Velocity in deep water is said to be :-
Square root(lambda*g/2*pie) where lambda is wavelength of the water wave and g is gravity.

However, when talking about ship hydrodynamics Froude number is defined to be V/square root(g*L) where L is waterline length of ship.

This implies that lambda/2*pie is taken to be equal to L
or, lambda = 2 * pie *L.

Does anyone know the justification for this.

Also, I would expect the value of V when FN is close to 1 to be related to planing speed, for hulls that are capable of planing but I cannot find any mention of this.
Thanks in anticipation
Dave Harrison

[DCockey]

The general definition of Froude number is a velocity divided by the square root of (the product of an acceleration, usually "g", and a length). Froude number is proportional to a ratio of velocity and the wave speed of gravity waves with wave length equal to the characteristic length. (This definition works when one fluid has a density which is much greater than the other, ie water and air.) The meaning of a particular numerical value of Froude number depends on the situation and characteristic length and velocity used.

Characteristic lengths used in naval architecture and hydrodynamics include (but not limited to):
- waterline length (This is the most common)
- water depth (Important when operating in shallow water, and can provide a measure of what shallow is. Wave speed varies with water depth.)
- cube root of the static immersed volume (Frequently used for planing.)
- static immersed transom depth (Used for flow and resistance with (static) immersed transoms.)

Confusion sometimes occurs when knowledge about shock and pressure waves associated with aircraft, missles, etc is used to try to understand water waves associated with a boat. Water/gravity waves are fundamentally different than pressure waves. The speed of propogation of gravity waves varies with wave length and the depth if the depth is not much greater than the wave length. Such waves are said to be dispersive. The wave pattern changes as the waves travel (expect for a few special cases).

The speed of propogation of pressure waves, including sound waves and shock waves, is independent of wave length or other lengths, and depends only on the fluid properties. Thus Mach number can be defined as the ratio of a characteristic velocity to the speed of sound without any reference to a characteristic length.

[dbharrison1]

Thanks DCockey
Thanks for the information but I am not sure it answers my question.

The propogation velocity of a gravity wave is :-
Square root(lambda*g/2*pie) ,where lambda is wavelength of the gravity wave.
So I would expect the Froude number to be defined as :-
V/Square Root(lambda*g/2*pie)

Taking the characteristic length to be L, the waterline length, this would give the Froude number as V/Square Root(L*g/2*pie) assuming wavelength is equal to the characteristic length.

However the Froude number is actually :
V/Square Root(L*g)

So the two differ by a factor Square Root(2*pie)

Hope this makes my problem clearer.
Dave Harrison

[daiquiri]

Dbharrison, though I'm not a native english speaker, please allow me to correct you on the spelling of the number 3.141592(...etc): it is called "pi", not "pie".

Froude number, as DCockey has mentioned, is a dimensionless number which expresses the ratio (inertial force) / (gravitational force) acting of the fluid element. The emphasize here is on the word "dimensionles". It doesn't depend on the unit system used to measure speed and length, which is very handy.
Now, when you multiply a pure (dimensionless) number by another pure number, the result is still a pure number. for example, you could multiply it by sq.root(2*pi) and it would still be a handy dimensionless number. But it would be a computational complication, and hence it is widely agreed that Fn is calculated in it's barest form:

Fn = V / sq.root(g*L)

As you have noted, the speed of a deep-water surface wave can be expressed indirectly, through its wave-length:

V = sq.root(l*g) / sq.root(2*pi) ; where l = wave length.

By substituting this expression into Fn definition, you obtain that

Fn=0.399*sq.root(l/L),

so when the wavelength is equal to the characteristic length of the (for example) boat, the Fn equals 0.399.
If we had defined the Fn as

Fn = sq.root(2*pi) * (V / sq.root(g*L)) ,

as you have written it, the only difference would be in the numerical value obtained. In that case instead of 0.399, the Fn would become 1 for a wavelength equal to L. It would surely be easier to remember, but nothing more.
So, resuming it in few words - it is all matter of accepted conventions.

Cheers!